Conic Sections: Circles, Ellipses, Parabolas & Hyperbolas

What Are Conic Sections?

Conic sections are the curves formed when a plane intersects a double-napped cone. The four types — circle, ellipse, parabola and hyperbola — are among the most important curves in mathematics, physics and engineering.

Learning Goals: By the end of this guide, you should be able to:

1. Write and recognise the standard equations of all four conics.
2. Find the centre, foci, vertices and asymptotes from an equation.
3. Understand eccentricity and how it classifies conics.
4. Convert between general and standard form.

The Four Conics

1. Circle

A circle is the set of all points equidistant from a centre $(h, k)$.

• Centre: $(h, k)$
• Radius: $r$
• Eccentricity: $e = 0$

2. Ellipse

An ellipse has two foci; the sum of distances from any point on the ellipse to the two foci is constant.

• Centre: $(h, k)$
• Semi-major axis: $a$, Semi-minor axis: $b$
• Foci: at distance $c = \sqrt{a^2 - b^2}$ from centre along the major axis
• Eccentricity: $e = \frac{c}{a}$, where $0 < e < 1$

3. Parabola

A parabola is the set of points equidistant from a focus and a directrix.

• Vertex: $(h, k)$
• Focus: at distance $|p|$ from vertex along the axis
• Directrix: line at distance $|p|$ on the other side
• Eccentricity: $e = 1$

4. Hyperbola

A hyperbola has two branches; the difference of distances from any point to the two foci is constant.

• Centre: $(h, k)$
• Vertices: at $(\pm a, 0)$ from centre
• Foci: at distance $c = \sqrt{a^2 + b^2}$ from centre
• Asymptotes: $y - k = \pm\frac{b}{a}(x - h)$
• Eccentricity: $e = \frac{c}{a}$, where $e > 1$

Eccentricity Summary

As eccentricity increases from 0, the curve "opens up" from a circle through increasingly elongated ellipses, then a parabola, then a hyperbola.

Worked Example

Question: Identify the conic $4x^2 + 9y^2 - 16x + 18y - 11 = 0$ and find its key features.

Step 1: Group and complete the square:

Step 2: This is an ellipse with centre $(2, -1)$, $a = 3$, $b = 2$, $c = \sqrt{9-4} = \sqrt{5}$.

Common Mistakes

1. Mixing up $a$ and $b$ in ellipse/hyperbola — $a$ is always the larger denominator for ellipses; for hyperbolas, $a^2$ is under the positive term.
2. Forgetting to complete the square fully — You must factor out the leading coefficient before completing the square.
3. Using the wrong formula for $c$ — Ellipse: $c^2 = a^2 - b^2$. Hyperbola: $c^2 = a^2 + b^2$. Notice the sign difference.

Exam Tips (A-Level / AP / IB)

• To quickly identify a conic from its general equation $Ax^2 + Bxy + Cy^2 + \ldots = 0$: if $A = C$ → circle; if $A$ and $C$ have the same sign but $A \neq C$ → ellipse; if one is zero → parabola; if opposite signs → hyperbola.
• Always sketch the conic after finding the key features — it helps catch algebraic errors.
• For parametric questions, the standard parametrisations are: circle $(r\cos t, r\sin t)$, ellipse $(a\cos t, b\sin t)$.

Frequently Asked Questions

Why is a circle considered a special case of an ellipse?

When $a = b$, the ellipse equation becomes $(x-h)^2 + (y-k)^2 = a^2$, which is a circle. The two foci merge into one point (the centre), and eccentricity drops to zero.

Where do conic sections appear in physics?

Planetary orbits are ellipses (Kepler's first law). Parabolas describe the path of projectiles. Hyperbolas appear in the orbits of objects with escape velocity and in radar/GPS triangulation.

Related Topics

• Function Transformations — Translations move conics without changing their shape.
• Spatial Geometry — Extending conic ideas to 3D quadric surfaces.
• Vector Operations — Directrix-focus problems involve distance vectors.